Complex Numbers (Book)

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Complex Numbers

Complex Numbers

A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.

The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.

Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.

Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.

http://www.ddewey.net/mandelbrot/

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10 Most Recent Blog Comments:

Write_Mikey_Write!

a day ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

a day ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

1 Like

Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

July 16, 2019 at 12:13am

#962697

Like Tears in Rain

Today, let’s talk movies. What was the most recent movie you watched? Have you been to the theater recently? What movie are you looking forward to? What is your favorite movie of all time?

Unless I'm traveling, not much gets me to leave the house these days.

I've even taken to having my groceries delivered. Lazy? Maybe, but there are other reasons. The big one is that if I go to the grocery store, there's always something to tempt me. Last time - I was there to fill a prescription - it was the Moon Landing 50th Anniversary Oreos.

Oreos are not conducive to weight loss efforts. But how could I resist? I couldn't, even though I'm not fond of marshmallow. If I'd never gone to the store, though, I never would have known about their existence.

I also discovered that the package has glow-in-the-dark lettering.

Bottom line is maybe I need to see about mail-order prescriptions. It's not like I have insurance to tell me that I must, or cannot, do so.

So, okay. Prescriptions, every other month or so, at least for now. The gym, every day, but it's not far. If friends are around, sometimes I'll go out for a beer. A massage once a month. The occasional grooming thing. Rare doctor appointments. The occasional miscellaneous errand, few and far between. And, a few times a year, a movie.

Finally, back on topic.

The last movie I saw - I think I talked about it here - was Spider-Man: Far From Home. Good movie. Fun. Might even go see it again. It just came out at the beginning of the month, and they had a 12:01 am showing on opening day that I had to go to because otherwise I'd encounter spoilers online.

Not just any movie can get me to leave the house. There's little point, for me, in watching dramas on the big screen. Superhero and other action movies? Definitely. Then, I can get invested. I'll watch all kinds of films on Amazon or Netflix, which don't require venturing into the not-so-great outdoors, but for the action movies, I'm happy to visit the theater. There's one a mile away from me that has draught beer. Score!

Missed out on the John Wick movie. I'll have to catch that on Amazon when it comes out there.

As for movies that I'm looking forward to, well, obviously, the final Star Wars movie of the trilogy of trilogies, which I believe comes out in December. I remember when Star Wars first came out, the buzz was that there were a total of 9 movies planned. It was touch and go there for a while, and forty-plus years is a long time to wait, but there's no way I'm not seeing it. There are probably ones I'll want to see before then; I just can't name any right now.

It's no secret that my favorite genres are fantasy and science fiction. Star Wars, for the record, is fantasy. It has science fiction props, sure, but it's not science fiction. Some call it "space opera." Whatever - genre is a marketing tool more than anything else; it helps us know what to expect. Comics-derived action movies exist in a fun little space between those two genres; I think that's part of their public appeal.

When it comes to choosing a favorite, though, one film stands above all others for me, and that is Blade Runner. The Director's Cut, not the silly version with narration.

It has its flaws, sure, just like everything else, but that particular movie speaks to me both as a writer and as a long-time science fiction reader.

The first true science fiction story - the one that started it all, the first story that actually attempted to use the real science of the time to drive the plot - was Mary Shelley's Frankenstein. I've known this for many years, now. We had to read it in college. (But wait a minute, Waltz, I thought you said you were an engineer! What is this "had to read" bullshit?) (I am. Was. Whatever. We were assigned it for - get this - Ethics in Engineering, fourth year.)

In the paper I had to write on that book (forget the movie; the movie had little to do with the book), I noted that the general distrust toward Frankenstein's creation was based on his appearance more than anything else. I vaguely recall a scene where it met a little girl, who ran screaming in terror, not for anything it did, or what it was (she couldn't know), but for what it looked like. If he'd made his creation pretty, it would have been accepted as just another human being. I proceeded to generalize this to science and invention in general. Got a C on the paper, as I recall; I was supposed to conclude that humans shouldn't "play God." I don't agree with that conclusion. What the hell; I passed, integrity intact.

Anyway, the point is, Blade Runner is the Frankenstein story turned on its head. In the case of that movie, the "creations," the replicants, are known to be replicants and distrusted based on that - no one could say that Rutger Hauer was ugly. At the time, that is; dude let himself go after that. I digress again. In Frankenstein, the creation turns on its creator and all of humanity; in Blade Runner, the creations turn on their creator, yes, but for good reason, and in the end (spoiler alert) the last one finds his humanity in the form of mercy and empathy.

In short, Frankenstein's creation was born good and learned evil; the replicants were born evil and learned compassion.

That's oversimplifying, of course, but I'm still calling Blade Runner my favorite all-time movie. Do not speak to me of the "sequel."

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